\documentclass[10pt,a4paper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}


\hyphenation{pro-per-ties}
\hyphenation{ge-ne-ral-ly}
\hyphenation{pre-fe-ren-ces}
\hyphenation{u-sing}
\hyphenation{pu-nish-ment}

\newcommand{\Pow}{\mathcal{P}}
\newcommand{\N}{\operatorname{N}}
\newcommand{\bool}{\operatorname*{\mathcal{B}}}
\newcommand{\Pos}{\operatorname{Pos}}
\newcommand{\Nec}{\operatorname{Nec}}
\newcommand{\T}{\mathcal{T}}


\author{Jose Enrique}
\title{Possibility and necessity for each Ill-known Constraint}
\begin{document}
Implementation of the ill-known constraints:
We will consider the following elements:
\begin{itemize}
\item $X$ an ill-known value.
\item $[D,c,e]$ a triangular possibility distribution. Where the core is $D$ and $D-c$ is the lower bound, $D+e$ is the higher bound.
\item $I = [a, b]$ a crisp interval where $a$ is the starting point and $b$ is the ending point.
\end{itemize}


\section*{$C\triangleq(\leq,X)$}
\begin{align}
\Pos (C([a,b])) &= \min_{r \in [a,b]} \sup_{\omega \leq r} \pi_X(\omega) = \\
\Pos (C([a,b])) &= \sup_{\omega \leq a} \pi_X(\omega)
\end{align}

So, for $I=[a,b]$ and  $X=[D,c,e]$ we have:

\begin{equation}
Pos (C[a,b]) = \mu_{\leq}(a) =
\begin{cases}
1 & \mbox{ if } a \leq D \\
\frac{a-(D+e)}{e} & \mbox{ if } a > D \wedge a < (D+e) \\
0 & \mbox{ otherwise }
\end{cases}
\end{equation}


\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/lte.pdf}
\end{figure}



\section*{$C\triangleq(<,X)$}
\begin{align}
\Pos (C([a,b])) &= \min_{r \in [a,b]} \sup_{\omega < r} \pi_X(\omega) = \\
\Pos (C([a,b])) &= \sup_{\omega < a} \pi_X(\omega)
\end{align}

\begin{equation}
Pos (C[a,b]) = \mu_{<}(a) =
\begin{cases}
1 & \mbox{ if } a < (D-c) \\
\frac{a-D}{c} & \mbox{ if } a \geq (D-c) \wedge a \leq D \\
0 & \mbox{ otherwise }
\end{cases}
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/lt.pdf}
\end{figure}


\section*{$C\triangleq(\geq,X)$}
\begin{align}
\Pos (C([a,b])) &= \min_{r \in [a,b]} \sup_{\omega \geq r} \pi_X(\omega) = \\
\Pos (C([a,b])) &= \sup_{\omega \geq b} \pi_X(\omega)
\end{align}
\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/gte.pdf}
\end{figure}

\begin{equation}
Pos (C[a,b]) = \mu_{\geq}(b) =
\begin{cases}
1 & \mbox{ if } b \geq D \\
\frac{(D-c)-b}{c} & \mbox{ if } a \geq (D-c) \wedge a \leq D \\
0 & \mbox{ otherwise }
\end{cases}
\end{equation}



\section*{$C\triangleq(<,X)$}
\begin{align}
\Pos (C([a,b])) &= \min_{r \in [a,b]} \sup_{\omega > r} \pi_X(\omega) = \\
\Pos (C([a,b])) &= \sup_{\omega > b} \pi_X(\omega)
\end{align}


\begin{equation}
Pos (C[a,b]) = \mu_{<}(b) =
\begin{cases}
1 & \mbox{ if } b \geq (D+e) \\
\frac{D-b}{e} & \mbox{ if } a \geq (D-c) \wedge a \leq D \\
0 & \mbox{ otherwise }
\end{cases}
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/gt.pdf}
\end{figure}


\section*{$C\triangleq(=,X)$}
\begin{align}
\Pos (C([a,b])) = \min_{r \in [a,b]} \sup_{\omega = r} \pi_X(\omega)
\end{align}
\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/eq.pdf}
\end{figure}

\section*{$C\triangleq(\neq,X)$}
\begin{align}
\Pos (C([a,b])) = \min_{r \in [a,b]} \sup_{\omega \neq r} \pi_X(\omega)
\end{align}
\begin{figure}[h]
\centering
\includegraphics[scale=1]{graphs/neq.pdf}
\end{figure}




\end{document}